Hey, guys.
Welcome to Algebra I.
Today's topic
will introduce you to
the world of polynomials.
You'll apply some of
what you learned in pre-algebra
to help you with this one.
Ready? Let's go.

(Describer) With a stylus, she activates a screen.

All right.
In order to know exactly
what a polynomial is,
I have to take you back
to what a monomial is.
A polynomial is the sum
of two or more monomials.
But what's a monomial?
Well, a monomial can be
just a regular number,
like 7 or 12 or a -4,
or the fancy way to say
a negative number in algebra
is a constant term
because it's value
doesn't change.
That 7 will constantly be 7.
It will never change.
That 12 is constantly 12,
and that -4 is constantly -4.
So, a monomial
can be a constant term,
or a regular number.
Okay?
A monomial
can also be a variable,
like x or v or y.
Those aren't constant terms
because their value can change,
so we call them variables.
A monomial could be a variable.
A monomial could also be
the product of a constant
and a variable.
For example,
5x or 7b-squared,
-12cd.
Those are also monomials.
Basically, a monomial,
in general, is one term.
Either a constant term,
a variable,
or the product of a constant
and a variable.
All of those
are considered monomials.
A binomial is a polynomial
that consists of two terms.
For example, 4x plus 1.
I've added two terms together.
It's called a binomial,
is its more specific name.
3b-squared minus 1?
That's a binomial,
X plus 3y.
That's a binomial.
A binomial is a polynomial
that only consists of two terms.
There's a lot of tongue-twisters
on this one.
Okay.
A trinomial--
I bet you might have noticed
from that prefix "tri"--
is a polynomial
with three terms.
For example,
5x-squared plus 2x plus 4.
Or, here we go,
4a-cubed b-squared
plus 6b minus 1.
That's a trinomial.
It's the sum of three terms.
Or x-cubed minus 8x plus 12.
That's a trinomial.
A trinomial is
a polynomial with three terms.
You can think of polynomial
as the name that encompasses
the big group.
But more specifically,
you could have a monomial,
which is just one term;
a binomial,
which is the sum of two terms;
or a trinomial,
which is the sum of three terms.
Collectively, we refer to them
as polynomials.
"Poly," that prefix--
it just means many.
Polynomial is
the sum of many terms.

(Describer) She changes the screen to show Example 1.

Okay.
One kind of problem
you'll run into
in the world
that is polynomials
is you have to find
the degree of the polynomial.
How you find
the degree of a polynomial
is you have to zone in
on the exponents.
Your largest exponent will be
the degree of the polynomial.
The value
of that largest exponent
is the value of the degree
of your polynomial.
We've got to go term by term
to see what exactly
is the degree of each term.
Bear with me for a second.
Get that pen.

(female describer)
5x-squared plus 3x-squared
y to the 4th power minus 6.

(Describer) 5x-squared plus 3x-squared y to the fourth power minus 6.

So, for 5x-squared,

(Describer) She writes.

my exponent
in that first term is a 2.
So, I can say the degree
of that first term is 2.
All righty?
Now, for the second term--
Yeah, let's go underneath.
3x-squared y to the 4th.
In that term,
I have two exponents.
When you have more than
one exponent in a term,
just add
your exponents together.
Here I have a 2 and a 4.
So, I'd say
the degree of this term
is 2 plus 4, which is 6.
Right?
So far, it's looking like
the largest degree
I've got going is 6.
Look at that -6,
that constant term.
We've got to throw back
to your knowledge
of exponents for a bit.
You remember that when you're
dealing with exponents,
for example, I can raise
anything to the 0 power
and it'll equal 1?
Right?
So, I could think about this
as -6 times x to the zero power,
because essentially it's
the same thing as -6 times 1.
I'm not changing the value
of that constant term.
But knowing that
and looking at it this way
will help you figure out
the degree
of that constant term.
Now you've introduced
an exponent to that term
without changing its value.
So, now I can see,
well, okay, the exponent 0.
So, the degree
of that constant term,
that -6, is 0.
Or a shortcut is that
you could think about it
like every constant term
will always have a degree of 0.
You can treat
every single constant term
in that same fashion.
For terms where you
actually see exponents,
just take them.
For 5x-squared,
the degree was 2.
For that 3x-squared
y to the 4th,
I had to add
those exponents together.
So, the degree was 6.
And then for my constant term,
that -6, the degree is 0.
Looking at all
of your degrees here,
you have a 2 right here,
you've got the 6,
and you have the 0.
Out of 2 and 6 and 0,
the biggest value is 6.
So, I'd say the degree
of my polynomial is 6.
Because it's whatever
that largest degree is
amongst all your terms.
Let's make sure
you've got the hang of that.
So, "What is the degree
of the polynomial?"
8x-squared z-cubed
plus 5z-squared minus 1.
Let's find the degree
of each of these terms.
Gotta shift that problem up,
give myself more room.
All righty.
For my first term,
I have 8x-squared z-cubed.
I have more than
one exponent here.
I need to add
the exponents together
to get the degree
of that term.
So, 2 plus 3, that's 5.
So, I'd say that first term
has a degree of 5. All righty?
Let's look at
that second term: 5z-squared.
I've only got one exponent
there, so that's simple.
The degree is 2.
My only exponent there is 2.
Then I see I have this -1 here.
That's a constant term.
You remember the shortcut
that we established
on that last one?
Every single constant term
has a degree of 0,
and we saw that
by looking at how
we could represent that product,
a constant term
times x to the zero power.
That constant term here, -1,
has a degree of 0.
Check out all your degrees.
Which one's the biggest?
The 5.
I'd say this polynomial
has a degree of 5.

(Describer) She writes "degree 5".

All right.
Let's keep going.
"What is the degree
of the polynomial?"
-7a-cubed b plus 2a-cubed
plus b-squared.
So, take each term separately,
each monomial separately.
All righty, so you have
-7a-cubed b.
You remember, with exponents,
when you don't see one--
like this term here,
or this part here, this b--
there's a 1 I'm not seeing.
I have a sum of exponents here,
3 plus 1.
I'll have to deal with that.
So, 3 plus 1, that's 4,
So, this monomial
has a degree of 4.
All righty.
Now I've got my 2a-cubed.
The 2a-cubed, I only see
this one exponent here.
That's simple.
This one has a degree of 3.
All righty.
Then that last monomial,
b-squared,
there's just one exponent
in this one.
So, the degree is 2.
Then check out
all your degrees.
You've got 4,
you've got 3, you've got 2.
The biggest one is 4.
So, the degree
of this polynomial...is 4.
All right.
Let's keep going.
And I do believe
it is your turn.
Press pause for a bit,
work through this problem,
and when you're ready to compare
answers with me, press play.

(Describer) Title: What is the degree of the polynomial?
9x to the fifth power y minus 4x cubed plus 2-squared.

(describer)
"What is the degree
of the polynomial?"
9x to the 5th power y
minus 4x-cubed plus x-squared.

(instructor)
Ready to check? Let's see.
Let's get the pointer.
Move it out of the way.
All righty.
9x to the 5th y
minus 4x-cubed plus x-squared.
What's the degree?
It is 6. Okay?
If you need to see how I
got that, this is how I got it.
Let's get the pen.
So, I took each term
individually.
So, 9x to the 5th y.
I didn't see
an exponent with that y.
That told me there was a 1
I wasn't seeing.
I have more than one exponent,
so I need to add them together.
So, 5 plus 1, that's 6.
I knew that first term
had a degree of 6.
Then I took
my next term, -4x-cubed.
I only had one exponent
to focus on there, the 3.
So, this term has a degree of 3.
Then I have the last term,
that x-squared.
Only one exponent to focus on,
and it's a 2.
So, your degree
of that term is 2.
Then check out all your degrees.
Which one's the biggest?
In this case, the 6.
So, the degree
of the polynomial was 6.
That's how I got that one. Okay?
Let's keep going.
A little more
to introduce you to polynomials.
You may also
run into problems
that ask you to find
the leading coefficient.

(Describer) Example 4:
8 x-squared minus two x plus seven x-cubed minus 4.

(describer)
Example 4:
8x-squared minus 2x
plus 7x-cubed minus 4.

(instructor)
How you do that is you
first have to make sure
that your polynomial
is written in standard form.
Standard form means
that your polynomial
goes from the term
with the largest exponent,
down to the term
with the smallest exponent.
That's how you write it
in standard form.
Sometimes the polynomial will be
given to you in standard form.
Then you just pick out
your leading coefficient.
But sometimes it's not
given to you in standard form.
You've got to do maneuvering
to get it there.
Looking at this polynomial,
I see it's not written
in standard form.
My largest exponent is a 3,
and that 7x-squared
is not written first.
I need to rearrange this.
All righty.
So, I'm gonna write this
as 7x-cubed,
'cause I'm gonna take
this first term first.
You notice how I'm circling
the plus sign in front of it?
That lets me know
what the sign of my term is.
I know this 7x-cubed
is a positive 7x-cubed,
and it has to be out front
if I write it in standard form,
'cause it has
the largest exponent.
So, 7x-cubed.
I've handled it,
so I'm gonna cross it out.
I've got an exponent of 3.
I see I have an exponent of 2,
this 8x-squared.
There's no sign out front.
You know from pre-algebra
that means it's positive. Okay.
So, I can write this
as 7x-cubed plus 8x-squared.
All righty?
And I've handled that.
I've got an exponent of 3,
exponent of 2,
and you have an exponent of 1.
That negative 2x,
there's a 1 there
that you're not seeing.
Gonna circle that term.
All righty.
You notice I circled
that subtraction out front?
That subtraction tells me
that 2x is negative.
When I'm rewriting
my polynomial,
I'm gonna write -2x.
Minus 2x, negative 2x.
They're kind of interchangeable.
You can think about them
the same way.
I've handled that.
I see I have this constant term
at the end, this -4.
So, at the end here: -4.
Now this polynomial
is written in standard form.
The exponents go from
3, 2, 1, and then a 0.
As you remember, I can think
about that -4 as a -4x to the 0.

(describer)
She's written 7x-cubed

(Describer) She's written 7x-cubed plus 8 x-squared minus 2x minus 4.

plus 8x-squared
minus 2x minus 4.
They don't always go sequential
like this,
an exponent of 3, an exponent
of 2, an exponent of 1.
Sometimes they skip one,
but you want to make sure
that those exponents go
in descending order.
Now that this is written
in standard form,
I can figure out
what the leading coefficient is.
The leading coefficient
is the coefficient
of your first term.
The leading term
is coefficient.
My first term is 7x-cubed,
and the coefficient there is 7.
So, the leading coefficient
of this polynomial is 7.
And you're all done.
It took a bit of work
to get there,
but the answer is
that constant, that 7.
Let's keep going.
What's the leading coefficient
of this polynomial?
I see I have 3x minus
x to the 4th plus 5x-squared.
I have to do some maneuvering,
because it's not
in standard form.
I know that
because I see that
that middle term
has the largest exponent, 4,
and it's not written first.
And when polynomials
are written in standard form,
it's gonna start
from the largest
and go down the line.
I need to do some rearranging.
That -x to the 4th
needs to be written first.
So, -x to the 4th.
All right, handled that.
I don't have an exponent of 3,
but I do have an exponent of 2.
It's attached to this term here.
That's a positive 5x-squared.
So, plus 5x-squared.
Handled that.
Then my last term is this 3x.
There's no sign
out front of it.
That tells me it's positive.
I can write that as plus 3x.
Now that this polynomial
is written in standard form,
now you can pick out
the leading coefficient.
Your first term is
-x to the 4th,
and same for coefficients
as exponents.
When you don't see
a coefficient,
there is
an invisible one there.
So, this first term
is like -1x to the 4th,
which would mean
that its coefficient is -1.
My leading coefficient
for this polynomial is -1.
Okay?
Let's try one more: 2 plus 9x.
All right.
That's not in standard form.
How do I know that?
Because that x
has an exponent of 1.
This 2 is a constant term.
I know there's
an x to the 0 there.
So, this isn't written
in the right order.
This term should be first.
It has the largest exponent.
So, 9x--It's positive, but you
don't have to write the plus
when it's the first term.
I've handled that.
Then you have the 2.
There's no sign out front,
so it's positive.
This really should be written
as 9x plus 2
for it to be
in standard form.
Now that it's
in standard form,
look at your leading term.
It's 9x, and the coefficient
of that first term is 9.
So, the leading coefficient
of that polynomial is 9.
All right?
All right, it's your turn.
Go ahead and press pause,
take a minute,
and work through these problems.
Remember how we did it.
First, make sure that polynomial
is written in standard form.
Then pick out
the leading coefficient.
When you're ready to compare
answers with me, press play.

(Describer) Title: What is the leading coefficient?
Number One: 3 plus 4x minus 9x-cubed plus 17x-squared
Number Two: 11 minus 3x-squared plus x to the fourth.

(describer)
What is
the leading coefficient?
Number 1: 3 plus 4x minus
9x-cubed plus 17x-squared.
Number 2: 11 minus 3x-squared
plus x to the fourth.
You ready to check?
Let's check.
Let's get the pointer tool.
For that first polynomial,
3 plus 4x minus 9x-cubed
plus 17x-squared,
the leading coefficient: -9.
For the second one, 11 minus
3x-squared plus x to the 4th,
that leading coefficient is 1.
If you need to see
how I did it, here we go.

(Describer) She switches to just the first problem.

So, first, I noticed
this wasn't written
in standard form,
because the term
with the largest exponent
wasn't written first.
I spotted that three, and I knew
it was a part of that term.
So, that should have been
written first: -9x-cubed.
All right.
Then I handled that.
I saw I had
an exponent of 2,
and that's the next one,
going in descending order.
That's a positive 17x-squared.
So, plus 17x-squared.
Now I handled that.
Now I see the 4x.
It's positive.
So, plus 4x.
All righty.
Then I see the 3.
There's no sign in front of it,
so it's positive.
So, plus 3.
Now that it's written
in standard form,
look at your first coefficient--
or look at your first term.
It's -9x-cubed,
and the coefficient
of that term is -9.
So, the leading coefficient
is -9.
Okay? Need to see
the second one?

(Describer) She switches to it.

All righty. 11 minus
3x-squared plus x to the 4th.
This wasn't written
in standard form,
so I had to rearrange it first.
That 4 is the largest exponent.
That means that x to the 4th
should be written first.
It's positive.
So, x to the 4th. Right?
Handled that.
Next exponent,
going in descending order,
would be 2,
and it's attached to that term.
So, -3x-squared.
Okay. Now I've handled that.
And that constant term, 11,
there's no sign in front of it,
so it's positive.
So, plus 11. Right?
Then I gave it a glance over.
I see that my first term
is x to the 4th.
I don't see a coefficient,
so I know
there's an invisible 1.
So, my leading coefficient is 1.
All right.
Hope you're getting comfortable
with the world of polynomials.
We'll do more with this topic,
but I hope that you got
a good introduction.
See you soon. Bye.

(Describer) Accessibility provided by the US Department of Education.